Serret-Frenet Formula 8.6 Equations of a Straight Line 8.7 Helix 8.8 Exercises 9 Intrinsic Geometry of Surfaces 9.1 Introduction 9.2 Curvilinear Coordinates on a Surface 9.3 Intrinsic Geometry: First Fundamental Quadratic Form 9.4 Angle Between Two Intersecting Curves on a Surface 9.5 Geodesic in Rn 9.6 Geodesic Coordinates 9.7 Parallel Vectors on a Surface 9.8 Isometric Surface 9.9 The Riemannian–Christoffel Tensor and Gaussian Curvature 9.10 The Geodesic Curvature 9.11 Exercises 10 Surfaces in Space 10.1 Introduction 10.2 The Tangent Vector 10.3 The Normal Line to the Surface 10.4 Tensor Derivatives 10.5 Second Fundamental Form of a Surface 10.6 The Integrability Condition 10.7 Formulas of Weingarten 10.8 Equations of Gauss and Codazzi 10.9 Mean and Total Curvatures of a Surface 10.10 Exercises 11 Curves on a Surface 11.1 Introduction 11.2 Curve on a Surface: Theorem of Meusnier 11.3 The Principal Curvatures of a Surface 11.4 Rodrigue’s Formula 11.5 Exercises 12 Curvature of Surface 12.1 Introduction 12.2 Surface of Positive and Negative Curvature 12.3 Parallel Surfaces 12.4 The Gauss-Bonnet Theorem 12.5 The n-Dimensional Manifolds 12.6 Hypersurfaces 12.7 Exercises
10 Part III: Analytical Mechanics 13 Classical Mechanics 13.1 Introduction 13.2 Newtonian Laws of Motion 13.3 Equations of Motion of Particles 13.4 Conservative Force Field 13.5 Lagrangean Equations of Motion 13.6 Applications of Lagrangean Equations 13.7 Himilton’s Principle 13.8 Principle of Least Action 13.9 Generalized Coordinates 13.10 Lagrangean Equations in Generalized Coordinates 13.11 Divergence Theorem, Green’s Theorem, Laplacian Operator, and Stoke’s Theorem in Tensor Notation 13.12 Hamilton’s Canonical Equations 13.13 Exercises 14 Newtonian Law of Gravitations 14.1 Introduction 14.2 Newtonian Laws of Gravitation 14.3 Theorem of Gauss 14.4 Poisson’s Equation 14.5 Solution of Poisson’s Equation 14.6 The Problem of Two Bodies 14.7 The Problem of Three Bodies 14.8 Exercises
11 Appendix A: Answers to Even-Numbered Exercises
12 References
13 Index
List of Illustrations
1 Chapter 7Figure 7.1Figure 7.2Figure 7.3Figure 7.4Figure 7.5Figure 7.6Figure 7.7
2 Chapter 8Figure 8.1Figure 8.2Figure 8.3Figure 8.4
3 Chapter 9Figure 9.1Figure 9.2Figure 9.3Figure 9.4Figure 9.5
4 Chapter 10Figure 10.1
5 Chapter 11Figure 11.1Figure 11.2
6 Chapter 12Figure 12.1Figure 12.2
7 Chapter 13Figure 13.1Figure 13.2Figure 13.3Figure 13.4Figure 13.5
8 Chapter 14Figure 14.1Figure 14.2Figure 14.3
Guide
1 Cover